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In other words, K[X] has the following universal property: For every ring R containing K, and every element a of R, there is a unique algebra homomorphism from K[X] to R that fixes K, and maps X to a. As for all universal properties, this defines the pair (K[X], X) up to a unique isomorphism, and can therefore be taken as a definition of K[X].
In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property. Technically, a universal property is defined in terms of categories and functors by means of a universal morphism (see § Formal definition , below).
The localization of a commutative ring R by a multiplicatively closed set S is a new ring whose elements are fractions with numerators in R and denominators in S.. If the ring is an integral domain the construction generalizes and follows closely that of the field of fractions, and, in particular, that of the rational numbers as the field of fractions of the integers.
For example, choosing a basis, a symmetric algebra satisfies the universal property and so is a polynomial ring. To give an example, let S be the ring of all functions from R to itself; the addition and the multiplication are those of functions. Let x be the identity function.
If R = Π i∈I R i is a product of rings, then for every i in I we have a surjective ring homomorphism p i : R → R i which projects the product on the i th coordinate. The product R together with the projections p i has the following universal property:
The field of fractions of is characterized by the following universal property: if h : R → F {\displaystyle h:R\to F} is an injective ring homomorphism from R {\displaystyle R} into a field F {\displaystyle F} , then there exists a unique ring homomorphism g : Frac ( R ) → F {\displaystyle g:\operatorname {Frac} (R)\to F} that extends h ...
Let R be a commutative ring and let A and B be R-algebras.Since A and B may both be regarded as R-modules, their tensor product. is also an R-module.The tensor product can be given the structure of a ring by defining the product on elements of the form a ⊗ b by [1] [2]
The same is true for a directed collection of subgroups of a given group, or a directed collection of subrings of a given ring, etc. The weak topology of a CW complex is defined as a direct limit. Let X {\displaystyle X} be any directed set with a greatest element m {\displaystyle m} .